- We have an exam on Friday following Spring Break
- Return of homework 4.7
- Nick - #7
- Robert - #16
- New assignment:
Fri 3/5 The end of dynamics; Fractals - No new reading in preparation for the next exam.
- Do pp. 450-, #4, 5, 13, 20 (just do two steps), Repeat 20 (but remove the center triangle that does not share a boundary with the original triangle; just do two steps), 21, 27, 32; at least one of these will figure somewhere on your next exam.
- Try my funhouse mirror fractal generator: make your own funhouse mirror image, using this web interface and your own image, and I'll post them on our "website gallery".
- We'll finish a problem from dynamics:
How many people do you suppose will be on the earth a hundred years from now?
- Start Somewhere (collect data!)
- Apply a process and get a result
- Apply the same process to that result to get a new result
- Apply the same process to the new result to get a newnew result
- Repeat patiently and persistently, forever.
- Fractals
- It's a graph that possesses "self-similarity".
- It often possesses a simple and recursive (or iterative) definition.
- We're going to be looking at more graphs in this unit, but they're
graphs of a very different nature.
- Let's start by having a glimpse at the beautiful gallery of
section 6.1, for these are the subjects of our unit.
- Deterministic
fractals: delve into an infinite, dynamic world, created by
a simple mathematical process:
- a simple rule
- repeated over and over and over.... but
- simple rules, intense content!
- Building deterministic fractals with sticks
- Let's see the process, but a little slower (check out the beautiful bushes...).
- Now let's introduce a little bit of randomness:
- Sierpinski triangle (p. 436):
- First deterministically: fractals by subtraction
- After we're finished removing all that stuff, how much area is left in the triangle?
- The Chaos game - generating fractals using random movement! (p. 447)
- Once again, simple rules lead to complex objects. This time, however, we don't proceed methodically, but rather haphazardly - and yet we still produce the complex images!
- Here's
a better version of that, that has more general options.
From randomness comes order; from simple rules comes complicated objects! Then all hell broke loose....
- Sierpinski triangle (p. 436):
- Mindscape #14, p. 452.
Here's a suitable mascot for this unit: Bessie the fractal cow: 
Have you ever lined up a couple of mirrors and seen an infinite number of copies of yourself disappearing off into the distance? Well, try my new funhouse mirror generator for fun (but no profit): make your own funhouse mirror image, using my web interface and your own image, and I'll post them on our "website gallery".
So what is a fractal? A couple of things we'll generally see:
